Monday, June 05, 2006

Poincare conjecture proved for real?

The Poincare conjecture is one the seven Clay millenium problems, each worth a million-dollar prize. It says something about all 3-dimensional "simply connected" manifolds being "equivalent" to the sphere in R4. A Fields medal was awarded for the proof for k-dimensional manifolds, k>4, and another Fields medal for k=4. (In those cases the "equivalence" is with the sphere in Rk+1.)

A few years ago, the Russian mathematician Grigori Perelman posted a series of papers on the arxiv. The papers outlined a proof of the "Thurston geometrization conjecture," a statement that implies the Poincare conjecture.

Ever since, the status of the Poincare conjecture has remained uncertain. Perelman's papers do not contain a full proof, but various experts have been in agreement that most likely Perelman's ideas can be successfully formalized. Meanwhile, Perelman's himself seemed to be oblivious to the matters of writing a full paper and claiming the prize.

Update 6/6/06: The Notices of the AMS have an article on the genesis of the Clay millenium prizes. The stories are very interesting. (For example Wiles's insights into mathematical politics.) I noticed one paragraph in particular. Perhaps with Perelman's case in mind, Arthur Jaffe writes

The rules for the prize resulted from a fair amount of thought. [...] One major safeguard involved the importance of publication of the solution. [...] Of course there can also be unforeseen circumstances. For example, an author of a solution may not write it down completely [...]

Update 6/7/06: An April 10 press releas from the organizers of ICM'06 suggests that there were plans to announce in Madrid that Perelman's proof has been "checked."

10 Comments:

Maybe they chose a minor journal because they were trying to scoop everyone else? There are other groups expounding upon and filling in Perelman's approach, most notably these guys. In any case, it is basically accepted amongst geometric analysts that Perelman's approach can be pushed through, and Perelman will soon get the Fields medal for his work.

The guardian article is idiotic, and completely misses the point that these guys are essentially writing expository notes about Perelman's work. It does seem like the Chinese fellows are attempting to take undeserved credit for proving the conjecture themselves.

...these guys are essentially writing expository notes about Perelman's work

Not clear. As the Xinhua article notes: "Perelman raised guidelines for proving the conjecture but [did] not specifically point out how to unravel the puzzle. 'Guidelines are totally different to complete proof of theories,' Yang said."

It is not even clear whether Perelman's writeup satisfies the requirements of the Clay prize (which is not to say he won't win a Fields medal for this work, regardless).

A quote from the recent preprint of Kleiner and Lott: "Regarding the proofs, [Perelman's papers] contain some incorrect statements and incomplete arguments, which we have attempted to point out to the reader... We did not find any serious problems, meaning problems that cannot be corrected by using the methods introduced by Perelman."

These are, I guess, two different statements of the same fact, with a vast difference in humility of the authors.

A quote from the recent preprint of Kleiner and Lott: "Regarding the proofs, [Perelman's papers] contain some incorrect statements and incomplete arguments, which we have attempted to point out to the reader... We did not find any serious problems, meaning problems that cannot be corrected by using the methods introduced by Perelman."

These are, I guess, two different statements of the same fact, with a vast difference in humility of the authors.

A quote from the recent preprint of Kleiner and Lott: "Regarding the proofs, [Perelman's papers] contain some incorrect statements and incomplete arguments, which we have attempted to point out to the reader... We did not find any serious problems, meaning problems that cannot be corrected by using the methods introduced by Perelman."

These are, I guess, two different statements of the same fact, with a vast difference in humility of the authors.

From an interview with Manuel de León (President of the Executive Committee of ICM2006): There’s no sign at the moment that [the Riemann hypothesis] can be resolved, and it’s important because...[of] its many practical implications in the technological field, such as data encryption.

Unbelievable that a distinguished(?) mathematician can make such a claim. =)

Talking about "scoop everyone else", please read the title of the article first: "A Complete Proof of the Poincaré and Geometrization Conjectures - application of the Hamilton-Perelman theory of the Ricci flow".

Does "application of the Hamilton-Perelman theory" not mean they give due credit to Perelman?

It is better to let the five Harvard professors to speak on this issue, since they had half-year exchange with the authors on this topic. And don't put any comments with racism or prejudice on this place.